On succinct convex greedy drawing of 3-connected plane graphs

Geometric routing by using virtual locations is an elegant way for solving network routing problems. In its simplest form, greedy routing, a message is simply for warded to a neighbor that is closer to the destination. It has been an open conjecture whether every 3-connected plane graph has a greedy drawing in <b>R</b>² (by Papadimitriou and Ratajczak [23]). Leighton and Moitra [20] recently settled this conjecture positively. One main drawback of this approach is that the coordinates of the virtual locations requires Ω(<i>n</i> log <i>n</i>) bits to represent (the same space usage as traditional routing table approaches). This makes greedy routing infeasible in applications. A similar result was obtained by Angelini et al. [2]. However, neither of the two papers give the time efficiency analysis of their algorithms. In addition, as pointed out in [16], the drawings in these two papers are not necessarily planar nor convex. In this paper, we show that the classical Schnyder drawing in <b>R</b>² of plane triangulations is greedy with respect to a simple natural metric function <i>H(u, v)</i> over <b>R</b>² that is equivalent to Euclidean metric <i>D</i>_<i>E</i><i>(u, v)</i> (in the sense that <i>D</i>_<i>E</i><i>(u, v) < H(u, v)</i> < 2√2<i>D</i>_<i>E</i><i>(u, v)</i>.) The drawing is succinct, using two integer coordinates between 0 and 2<i>n</i> − 5. For 3-connected plane graphs, there is another conjecture by Papadimitriou and Ratajczak (as stated in [16]): Convex Greedy Embedding Conjecture: Every 3-connected planar graph has a convex greedy embedding in the Euclidean plane. In a recent paper [6], Cao et al. provided a plane graph <i>G</i> and showed that any convex greedy embedding of <i>G</i> in Euclidean plane must use Ω(<i>n</i>)-bit coordinates Thus, if we add the succinctness requirement, the Convex Greedy Embedding Conjecture is false. In this paper, we show that the classical Schnyde drawing in <b>R</b>² of 3-connected plane graphs is <i>weakly greedy</i> with respect to the same metric function <i>H</i>(*, *). The drawing is planar, convex, and succinct, using two integer coordinates between 0 and <i>f</i> (where <i>f</i> is the number of internal faces of <i>G</i>).